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Properties

Label 20.4.40487532402...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 331\cdot 691^{4}\cdot 41351$
Root discriminant $42.69$
Ramified primes $3, 5, 331, 691, 41351$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 40, -25, -270, 964, -1657, 1688, -960, 141, -130, 1094, -2360, 3080, -2906, 2120, -1224, 561, -201, 54, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 54*x^18 - 201*x^17 + 561*x^16 - 1224*x^15 + 2120*x^14 - 2906*x^13 + 3080*x^12 - 2360*x^11 + 1094*x^10 - 130*x^9 + 141*x^8 - 960*x^7 + 1688*x^6 - 1657*x^5 + 964*x^4 - 270*x^3 - 25*x^2 + 40*x + 5)
 
gp: K = bnfinit(x^20 - 10*x^19 + 54*x^18 - 201*x^17 + 561*x^16 - 1224*x^15 + 2120*x^14 - 2906*x^13 + 3080*x^12 - 2360*x^11 + 1094*x^10 - 130*x^9 + 141*x^8 - 960*x^7 + 1688*x^6 - 1657*x^5 + 964*x^4 - 270*x^3 - 25*x^2 + 40*x + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 54 x^{18} - 201 x^{17} + 561 x^{16} - 1224 x^{15} + 2120 x^{14} - 2906 x^{13} + 3080 x^{12} - 2360 x^{11} + 1094 x^{10} - 130 x^{9} + 141 x^{8} - 960 x^{7} + 1688 x^{6} - 1657 x^{5} + 964 x^{4} - 270 x^{3} - 25 x^{2} + 40 x + 5 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(404875324028746129193940673828125=3^{12}\cdot 5^{12}\cdot 331\cdot 691^{4}\cdot 41351\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.69$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 331, 691, 41351$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{18752} a^{18} - \frac{9}{18752} a^{17} - \frac{1823}{9376} a^{16} - \frac{2033}{4688} a^{15} + \frac{4631}{18752} a^{14} - \frac{3333}{18752} a^{13} + \frac{3795}{9376} a^{12} + \frac{5475}{18752} a^{11} + \frac{9353}{18752} a^{10} - \frac{333}{1172} a^{9} - \frac{3725}{18752} a^{8} - \frac{9055}{18752} a^{7} - \frac{5155}{18752} a^{6} - \frac{87}{9376} a^{5} - \frac{4661}{18752} a^{4} - \frac{413}{4688} a^{3} + \frac{7479}{18752} a^{2} - \frac{8411}{18752} a + \frac{8271}{18752}$, $\frac{1}{18752} a^{19} - \frac{3727}{18752} a^{17} - \frac{1721}{9376} a^{16} + \frac{6451}{18752} a^{15} + \frac{421}{9376} a^{14} - \frac{3655}{18752} a^{13} - \frac{1223}{18752} a^{12} + \frac{593}{4688} a^{11} + \frac{3841}{18752} a^{10} + \frac{4579}{18752} a^{9} - \frac{1269}{4688} a^{8} + \frac{3555}{9376} a^{7} - \frac{9065}{18752} a^{6} - \frac{6227}{18752} a^{5} - \frac{6097}{18752} a^{4} - \frac{7389}{18752} a^{3} + \frac{661}{4688} a^{2} + \frac{1895}{4688} a - \frac{569}{18752}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 559646808.525 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.5438807015625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $16{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.12.12.6$x^{12} + 24 x^{11} - 3 x^{10} + 81 x^{9} - 18 x^{8} + 54 x^{7} + 108 x^{5} - 54 x^{4} - 27 x^{3} - 81 x - 81$$3$$4$$12$12T39$[3/2, 3/2]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
331Data not computed
691Data not computed
41351Data not computed